Divide the following complex numbers: $\dfrac{8 e^{\pi i}}{4 e^{19\pi i / 12}}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Solution: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $8 e^{\pi i}$ ) has angle $\pi$ and radius 8. The second number ( $4 e^{19\pi i / 12}$ ) has angle $\frac{19}{12}\pi$ and radius 4. The radius of the result will be $\frac{8}{4}$ , which is 2. The difference of the angles is $\pi - \frac{19}{12}\pi = -\frac{7}{12}\pi$ The angle $-\frac{7}{12}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{7}{12}\pi + 2 \pi = \frac{17}{12}\pi$ The radius of the result is $2$ and the angle of the result is $\frac{17}{12}\pi$.